Checking method for calculating free charge from Electric field in LIH

[tomwilliam]

Homework Statement

I'm asked to find the free charge per unit length on a cylinder which is surrounded by a LIH dielectric material. I have an expression for the electrostatic potential $$V(r)$$ in cylindrical coordinates.

Homework Equations

$$\mathbf{D}=\varepsilon \varepsilon_0 \mathbf{E}$$
Gauss's Law in media equating flux of displacement field across a surface to the free charge.
$$\mathbf{E}=-grad V$$

The Attempt at a Solution

I've calculated E using the last equation there, then used it (and the LIH assumption) to calculate D, and now I can equate:
$$D_r \times 2 \pi r L = Q_f$$
Where L is the length of the cylinder and Q_f is the free charge.
This is enough to give me the expression for free charge per unit length. My problem is: am I right in taking the value of E at the cylinder surface?
I know that the potential is zero on this surface, so that seems to throw my working out a little.
Any advice appreciated.

 

[Gordianus]

Why do you think the potential should be zero on the surface?

 

[tomwilliam]

I know that the potential is zero at the surface because I have the expression for V(r) and it works out as zero on the surface.

I've just realized, though, that I could choose an arbitrary Gaussian surface in the dielectric and use that to calculate the free charge per unit length.

Would that work?